Functions and Graphs

Linear Equations

A linear equation (a straight line) has the form: y = mx + c, where m is the slope, and c is the displacement in the y-axis.

Two straight lines are either parallel or they will cross at one point.

How can you find the point of intersection of two lines? You could graph the lines to see the point of intersection, but you can also use the equations:

Graphical Method

Supposing we have two lines: f(x) = 4x - 2, and f(x) = -3x + 6.

We plot them both on the one axis:

2 lines that are not parallel will intersect at one point

The exact point of intersection is not so easy to determine from a graph!

Algebraic Method: simultaneous equations

Theory: there is one, and only one, point where the values of x and y are the same for both lines. At that point the two lines are identical.

Let f(x) = y. We will use the simultaneous equations method to solve for x. Then we will put this value back into one of the equations to find y. Remember, since both lines are identical at this point, it does not matter which equation we use (try them both).

$$y = 4x - 2$$
$$y = -3x + 6$$

Subtract one equation from the other: be careful that when you subtract a negative number, it is the same as adding a positive number.

$$0 = 7x - 8$$

So,

$$7x = 8$$
$$x = {8/7}$$

Putting this value back into either of the equations to find y:

$$y = 4x - 2 = 4⋅(8/7) - 2 = {32}/7 - 2 = {32 - 14}/7 = {18}/7$$

The point where the two lines intersect is (${8/7}$, ${18}/7$).

Function of functions

Sometimes it is easier to solve a function if it is made into two functions.

Inverse functions

To find the inverse of a function, we write y in place of f(x), and reverse the places of x and y:

$x = y^2 + 1$

$y^2 = x - 1$, so $y = ±√{x - 1}$

Line of Best fit

When data is collected it usually creates a scatter pattern of points when plotted on a graph. This may be due to natural variations in the data set, but it may also be due to errors and uncertainty in the measurement techniques.

In order to see the general pattern, and from this derive a general relationship between the quantity graphed as x and the quantity graphed as y, a line of best fit may be drawn.

A line of best fit is a kind of average. It is a line (or curve) which seems to pass through the middle of the data points, so that there is as much variation to the line above as below.

Correlation

The correlation of two data sets can be seen from a line of best fit. If the line is sloping upwards, from left to right, it is a positive correlation. This means that as x increases, so does y. For example, the income of people has a positive correlation with the size of house they live in.

A negative correlation is one which has a negative or inverse relationship between two quantities. The example here is of test results for children in two subjects: French and Maths. As the scores increase for French, they decrease for Maths. Not always, there are exceptions, but in general, children who like Maths tend not to do so well in languages, according to this data set.

Lines of best fit allow the mean (66.5, 52.5 in the example) to be determined quickly.

If one of the variables is known (Maths = 70 in the example), a value for the other variable (French = 52.5% in the example) can be determined easily from the line of best fit. This represents the most likely value for this variable. The true value could vary from this as much as the other points vary on average. There are statistical methods to determine this expected variation, known as standard deviation.

Rotational Symmetry

Rotational symmetry is the number of identical shapes or graphs created when an object is rotated through 360°. An object which has no symmetry (such as a non-symmetrical polygon) has order of rotational symmetry 1.

A straight line has an order of rotational symmetry of 2. A parabola has an order of rotational symmetry of 1.